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<title>Simulations for Statistical and Thermal Physics</title>

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<h3 style="text-align:center;">Geometric phase transition</h3>

<p class="header_title">Introduction</p>

<p>The goal of this program is to demonstrate some of the important properties of percolation, especially near the geometrical phase transition. The key idea is the non-existence 
of a spanning path for p &lt; p<sub>c</sub> and the existence of a spanning path for p &#8805; p<sub>c</sub>, where p<sub>c</sub> is the percolation threshold.</p>

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<p class="header_title">Algorithm</p>
<ol>
<li>An empty lattice is generated and sites are occupied with probability p. (The occupancy of a given site is independent of the occupancy of all other sites.)</li>

<li>The Newman-Ziff algorithm is used to identify the clusters.</li>

<li>Cluster properties are computed as a function of p. They include the probability of a site being in the spanning cluster, 
the probability that there is a spanning cluster, and the mean cluster size.</li>

<li>The cluster size distribution is computed at p = p<sub>c</sub> &#8773; 0.5927, the percolation threshold for the square lattice.</li>

<li>Repeat steps 1&#8211;4 and average over the trials and plot results.</li>

</ol>

<p>&nbsp;&nbsp;&nbsp;&nbsp;A key property of systems undergoing a continuous phase transition is that 
there exist clusters (such as the clusters of occupied sites in site percolation) of all sizes at the transition. 
For p &#8800; p<sub>c</sub> there are only small clusters in the nonspanning ("disordered") phase and only 
the one spanning cluster and small clusters in the "ordered" phase. The small clusters can be characterized by a typical 
length (called the correlation length in thermal systems and the connectedness length in percolation problems) which diverges
as a power law as the transition is approached. Other quantities also follow a power law near the transition.</p>

<p class="header_title">Problems</p>

<ol>

<li>Run the program and look at the percolation configurations. The program only shows configurations near the percolation threshold,
p<sub>c</sub> &#8773; 0.5927. Note that clusters appear at many different sizes. Because the system is close to the transition, there is sometimes a 
spanning cluster and sometimes not. Increase the lattice linear size to 256 or greater and qualitatively describe the distribution of cluster sizes.</li>

<li>Run the program again and after about 100 trials stop the program and look at the cluster size distribution (a log-log plot). Do you
 see a linear plot for at least part of the data? What functional form does this linear dependence imply? Use the <tt>Data Table</tt> menu item under
<tt>Views</tt> and copy the data into a plotting or spreadsheet program and estimate the slope of the linear part of the data. The exact result for this
slope for an infinite system is -&#964; = -187/91 &#8773; 2.05. How does your result compare?</li>

<li>A spanning cluster is defined in the program to be one that touches both the left and right edge of the lattice. As soon as a cluster spans horizontally
we say there is a spanning cluster. In the infinite lattice limit clusters that span horizontally will always span vertically. For L = 128 estimate the value 
of p<sub>c</sub> as the value of p where half the trials span. How does your estimate compare with 0.5927? How would you expect your estimate to change if 
the spanning rule was that a cluster had to span both horizontally and vertically to be labeled a spanning cluster? How would you expect your estimate to change if 
the spanning rule was that a cluster that spanned either horizontally or vertically?</li>

<li>Choose L = 128 and do at least 100 trials (1000 is better). Copy the data for the mean cluster size and the probability P<sub>&#8734;</sub> that an occupied site is in the spanning cluster (P<sub>&#8734;</sub> is called the order parameter). 
Make a log-log plot of P<sub>&#8734;</sub> versus |p-p<sub>c</sub>| with p<sub>c</sub> &#8773; 0.5927. There should be a region of 
your plot that is linear on this log-log graph, indicating a power law. Estimate the slopes. For the mean cluster size the exact result for an infinite lattice 
is -&#947; = -43/18 and
for the order parameter it is &#946; = 5/36.</li>

<li>It is likely that your results for the slopes in Problem 4 are not very accurate. Another approach, called finite size scaling, is to recognize that the correlation
length also follows a power law with an exponent &#957; = 4/3. We can replace |p-p<sub>c</sub>| by L<sup>-1/&#957;</sup>, and log-log plots of the
order parameter versus L at the transition should be linear with a slope of -&#946;/&#957;. Similarly, the mean cluster size should scale as L<sup>&#947;/&#957;</sup>. Use the program to compute
&#946;/&#957; and &#947;/&#957; for L = 10, 20, 40, 80, and 160, and compare with the exact results for &#946; and &#947;. Use the exact result &#957; = 4/3. Because &#946; is so small, your results for &#946;/&#957; are likely to have
a large percentage error.</li>

</ol>

<p class="header_title">References</p>

<ul>

<li>Dietrich Stauffer and Ammon Aharony, <i>Introduction To Percolation Theory,</i>
Taylor &#38; Francis (1985).</li>

<li>H. Gould, J. Tobochnik, and W. Christian, <i>An Introduction to Computer Simulation Methods</i> (Addison-Wesley, 2006), 3rd ed., Chapter 12.</li>
</ul>

<p class="header_title">Java Classes</p>

<ul>

<li>ClustersApp</li>
<li>Clusters</li>

</ul>


<p class = "small">Updated 27 February 2007.</p>
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